I installed Keepass 1.16 in my thumbdrive, which reportedly works with wine (as per Keepass homepage), but after entering the master password it just hangs and shows the "The program keepass has encountered a serious problem and needs to close" screen.

If you're looking to use the Keepass 1.x version, you should just use KeepassX instead.

It's as easy as adding the KeepassX repository to your sources (via System->Administration->Software Sources : Third Party Software->Add), and then installing the package.

sudo apt-get install keepassx

Note that if you're using Keepass 2.x versions on Windows that use the .kdbx file extensions, you're out of luck as even the latest KeepassX 0.4.0 Beta 1 version is not able to work with this. I stumbled across this issue and had to revert to using a Keepass 1.x version on Windows for compatibility.

Edit: As Decio pointed out in the comments, Keepass 2.x runs on the .NET framework, hence can be run under Mono, albeit using a Mono version that's not in the Ubuntu repository at this time.